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midterm3-A-solution-public

# midterm3-A-solution-public - Math 201 Fall 2011 Midterm...

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Math 201 Fall 2011 Midterm Exam 3 (A) November 29, 2011 1. [5] Set up (but do not solve) the differential equation of motion for a spring-mass system where the mass is 2 units, the damping coefficient is 3 and the spring constant is 4 with an external applied force F ( t ) = 0 , 0 t < 2 , 7 , 2 t < 3 , 0 , t > 3 , using the unit step function; The equation is mx 00 + γx 0 + kx = F ( t ) , where the mass m = 2 , the spring constant k = 4 , the damping coefficient γ = 3 . F ( t ) = 7[ U ( t - 2) - U ( t - 3)] . So, the equation is 2 x 00 + 3 x 0 + 4 x = 7 U ( t - 2) - 7 U ( t - 3) . 2. [5] Find the general solution of x 2 y 00 + 5 xy 0 + 4 y = 0 . The auxiliary equation is m ( m - 1)+5 m +4 = m 2 +4 m +4 = ( m +2) 2 = 0 , a repeated root m = - 2 . Thus the two linearly independent solutions are y 1 = x - 2 and y 2 = y 1 ln x = x - 2 ln x . The general solution is thus y = c 1 y 1 + c 2 y 2 = c 1 x 2 + c 2 ln x x 2 3. [10] Use Laplace transforms to solve dy dt + y = e - t sin t , y (0) = 0 . Take Laplace transforms on both sides, let Y = L{ y } , - y (0) + sY + Y = L{ e - t sin t } = 1 ( s + 1) 2 + 1 Substitute y (0) = 0 , and isolate Y , Y = 1 ( s + 1)[( s + 1) 2 + 1] 1

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midterm3-A-solution-public - Math 201 Fall 2011 Midterm...

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